How could I design a paper spool or drum that is moving at constant speed without electrical system? What mechanisms are available?
I know that some paper tracing system was being used in 19th century to track vibration frequency.
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There are a few ways to go about this. One easily available solution is simply to adapt the mechanism from a mechanical clock or watch. This will give a regular series of pulses rather than a true constant speed but in many cases this will not be a problem and may even be an advantage.
Another approach is to use a regulator or governor to achieve a constant speed, either directly or via a mechanical control system.
On way to achieve this, for example is to have a heavy weight turning a pulley with a large reduction ratio and coupled to some sort of damper (like a fan). In this case acceleration due to gravity and aerodynamic drag on the fan will quickly reach a stable equilibrium.
Yet another method is to have an approximately constant speed drive and a mechanism which regularly punches or otherwise marks the paper roll so you have a regular 'tick' as a baseline for comparison. For example a pendulum is a simple and effective way of achieving a regular and repeatable time interval.
If you are trying to simulate 19th century technology, based on your other question, you could achieve this to a pretty good precision with a calibrated metronome and a skilled operator turning a crank. That could easily provide enough precision for your vibration measurement experiments.
Things don't need to be any more complicated than they have to.